A Zipf discrete random variable.
Discrete random variables are defined from a standard form and may require some shape parameters to complete its specification. Any optional keyword parameters can be passed to the methods of the RV object as given below:
Parameters: | x : array_like
q : array_like
a : array_like
loc : array_like, optional
size : int or tuple of ints, optional
moments : str, optional
Alternatively, the object may be called (as a function) to fix the shape and location parameters returning a “frozen” discrete RV object: rv = zipf(a, loc=0)
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Notes
The probability mass function for zipf is:
zipf.pmf(k, a) = 1/(zeta(a) * k**a)
for k >= 1.
zipf takes a as shape parameter.
Examples
>>> from scipy.stats import zipf
>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots(1, 1)
Calculate a few first moments:
>>> a = 6.5
>>> mean, var, skew, kurt = zipf.stats(a, moments='mvsk')
Display the probability mass function (pmf):
>>> x = np.arange(zipf.ppf(0.01, a),
... zipf.ppf(0.99, a))
>>> ax.plot(x, zipf.pmf(x, a), 'bo', ms=8, label='zipf pmf')
>>> ax.vlines(x, 0, zipf.pmf(x, a), colors='b', lw=5, alpha=0.5)
Alternatively, freeze the distribution and display the frozen pmf:
>>> rv = zipf(a)
>>> ax.vlines(x, 0, rv.pmf(x), colors='k', linestyles='-', lw=1,
... label='frozen pmf')
>>> ax.legend(loc='best', frameon=False)
>>> plt.show()
Check accuracy of cdf and ppf:
>>> prob = zipf.cdf(x, a)
>>> np.allclose(x, zipf.ppf(prob, a))
True
Generate random numbers:
>>> r = zipf.rvs(a, size=1000)
Methods
rvs(a, loc=0, size=1) | Random variates. |
pmf(x, a, loc=0) | Probability mass function. |
logpmf(x, a, loc=0) | Log of the probability mass function. |
cdf(x, a, loc=0) | Cumulative density function. |
logcdf(x, a, loc=0) | Log of the cumulative density function. |
sf(x, a, loc=0) | Survival function (1-cdf — sometimes more accurate). |
logsf(x, a, loc=0) | Log of the survival function. |
ppf(q, a, loc=0) | Percent point function (inverse of cdf — percentiles). |
isf(q, a, loc=0) | Inverse survival function (inverse of sf). |
stats(a, loc=0, moments=’mv’) | Mean(‘m’), variance(‘v’), skew(‘s’), and/or kurtosis(‘k’). |
entropy(a, loc=0) | (Differential) entropy of the RV. |
expect(func, a, loc=0, lb=None, ub=None, conditional=False) | Expected value of a function (of one argument) with respect to the distribution. |
median(a, loc=0) | Median of the distribution. |
mean(a, loc=0) | Mean of the distribution. |
var(a, loc=0) | Variance of the distribution. |
std(a, loc=0) | Standard deviation of the distribution. |
interval(alpha, a, loc=0) | Endpoints of the range that contains alpha percent of the distribution |